We study the field structure and dispersion properties of a hybrid eigenmode guided by a nonuniform magnetized plasma waveguide. It is shown that the rotational and quasi-potential waves contribute to the formation of such a mode in the whistler frequency range. Depending on the plasma density, the rotational component of the hybrid mode is determined by either waves with complex transverse wave numbers or whistler waves, or by true surface waves. In the presence of an axial nonuniformity of the plasma in a channel, the transverse field structure of the propagating mode changes, which is stipulated by changes in both the values of transverse wave numbers and their dependence on the radial coordinate. It is found that the spectrum of axial wave numbers of eigenmodes of a plasma waveguide undergoes a pronounced condensation when smoothing the waveguide walls. The damping of the hybrid mode of a nonuniform waveguide due to electron collisions is found and it is shown that collisional losses determine the damping of waves trapped in the waveguide in the experiments on ionization self-channeling of whistler waves. We have found the effect of "displacing" the strong field from the inner core to the background outer region of the waveguide with increasing plasma density on its axis and broadening background region.